Probing the structural, magneto-electronic, thermophysical, and thermoelectric properties of vanadium-based V2MnZ (Z = As, Ga) Heusler alloy: a computational assessment

The structural, magneto-electronic, thermophysical and thermoelectric properties of vanadium-based V2MnZ (Z = As, Ga) alloys have been investigated using density functional theory simulation scheme and semiclassical Boltzmann transport methods. First of all, the structural characterization has been performed in ferromagnetic and non-magnetic states which signifies that both the alloys crystallized in C1b type structure with space group F-43m. We also computed the various thermodynamical parameters such as heat capacity (Cv), Debye temperature (θ D), and grüneisen parameter (γ) of these materials, with the help of quasi-harmonic approximation (QHA) at pressures ranging from 0 to 20 GPa and temperatures ranging from 0 to 900 K. We have used the Boltzmann transport theory with the constant relaxation approximation as a basis for calculating various thermoelectric coefficients such as the Seebeck coefficient, power factor, total thermal conductivity and figure of merit. The efficient half-metallicity and thermoelectric responses contribute to spintronics and green energy harvesting technology.


Introduction
In the field of research, there is a significant interest in discovering new materials and phenomena that can be used in energy-efficient sensors and memories.Recent studies have focused on identifying new ferromagnetic materials and alternative compounds, such as half-metallic ferromagnets, which have great potential for technological applications.The use of spin instead of charge to communicate information in spintronic devices [1] provides several advantages, including low energy consumption and nonvolatility for magnetic randomaccess memories (MRAM), spin valves, giant magneto resistances [2], and other devices.Heusler alloys [3,4] have received extensive attention due to their excellent chemical and physical properties, including high spinpolarization, interesting electronic structure, shape memory, high Curie temperatures, ferromagnetism, and thermoelectric properties.They are used in various applications, including magneto-resistive materials, spin valve generators, spin filters, transducers, shape memory devices, and spintronics [5][6][7][8].Vanadium-based full Heusler alloys [9] possess half-metallic ferromagnets (HMFs) behavior, while half-Heusler alloys behave differently.The discovery of new materials with half-metallic behavior and 100% spin polarization has led to their use in spin injectors for magnetic random-access memories and other spin-dependent devices [10].
Various Co-based, Mn-based, Fe-based and V-based half-metallic compounds have been investigated widely and most of the predicted compounds follow the S-P rule [11,12] very well and exhibit half-metallic properties.Sofi et al have worked on Co 2 PAl (P = Ru, Rh) Heusler alloy [11] to illustrate the application in research fields like energy harvesting and thermoelectric technology.Guermit et al have reported Mn 2 RhZ (Z = Al, Si and Ge) [12] to explore the potential of these alloys in technological devices.Rached et al [13] worked on XCrSb (X = Fe, Ni) alloys and concluded that they are the potential candidates for spintronics.The main aim of our paper to investigate how the family of Heusler compounds can be modified to produce novel half-metallic magnets with high Curie temperatures, thereby increasing the materials database for applications in spintronics.To the best of our knowledge, there have been site preference, half-metallicity and electronic properties have been determined.on this compound.This paper investigates the magneto-electric, structural, thermodynamic, and transport properties of V 2 MnZ (Z = As, Ga) inverse [14,15] Heusler alloys and discusses their halfmetallicity.The electronic structure of these materials has been studied using first-principles calculations.

Computational details
Density functional theory (DFT) is a reliable method for assessing the compound's ground state description in order to fully comprehend its physical properties.Highly accurate ground state parameters of the current materials are calculated using DFT calculations within the framework of the full potential linearized augmented plane wave (FP-LAPW) approach as implemented in WIEN2k [16,17].The Kohn-Sham equation is solved to determine the ground state properties.PBE-GGA effectively determines the ground state characteristics, including bulk modulus, energy, and lattice constant, however it underestimates band gaps and electronic properties.Therefore, in order to obtain the precise and accurate bands in the current system, we have adopted the modified Becke Johnson (mBJ) method.A cut-off parameter of R mt K max = 7 is employed, where R mt is the smallest muffin tin radii and K max is the magnitude of the largest k-vector.To ensure better convergence, 1500 k-mesh points are used for self-consistent calculations.The thermoelectric properties are calculated using the BoltzTraP code [18] under the assumption of constant relaxation time.The Gibbs2 code [19] evaluates some thermodynamic parameters, such as the Debye temperature and Grüneisen parameters.

Results and discussion
This section reveals the analysis of various properties of the alloys.

Structural properties
Heusler alloys are also known as intermetallic or smart metals.They are ternary intermetallic compounds.Heusler is classified into three groups: Full Heusler alloys, Half Heusler alloys, and Quaternary Heusler alloys [20].Based on stoichiometric composition, the general formula of full Heusler alloys is X 2 YZ having a stoichiometric ratio of 2:1:1, crystallized in the L2 1 structural phase.In contrast, half-Heusler alloys with the general formula XYZ have a stoichiometric ratio of 1:1:1 and crystallize in C1 b phase.Quaternary Heusler alloys have XX'YZ, which crystallize in the C1 b phase, where X and Y are transition elements, and Z is p-block elements [21].For the current investigation, we have, Vanadium based Heusler alloys which crystallize in the C1 b phase with space group F-43m (#216).Figure 1 shows the crystal structure of V 2 MnZ (Z = As, Ga).The Wyckoff's position, for the current set of V 2 MnZ (Z = As, Ga) alloys, occupied the following positions V-I (0, 0, 0), V-II (0.25, 0.25, 0.25), Mn (0.50, 0.50, 0.50), and Z (As, Ga) (0.75, 0.75, 0.75) [22].The general formula for regular Heusler alloys is X 2 YZ, and inverse Heusler alloys are Y 2 XZ.Suppose the X atoms have more valence electrons than the Y atoms.In that case, the compound is said to be a regular Heusler alloy that crystallizes in the Cu 2 MnAl-prototype and L2 1 structure with the Fm-3m space group.At the same time, if the Y atom has more valence electrons than the X atom, the compound is said to be inverse Heusler alloys [23,24] which crystallizes in Hg 2 CuTi-prototype and C1 b structure with F-43m space group.Vanadium-based inverse Heusler alloys are stable in the Hg 2 CuTi prototype with a C1 b structure.The compounds are optimized in both ferromagnetic (FM) and non-magnetic (NM) states and show stability in the FM state.The optimized lattice constant for V 2 MnAs is 0.58 nm and for V 2 MnGa is 0. 59 nm.Figures 2 and 3 shows the energy versus volume curves in FM and NM states for V 2 MnZ (Z = As, Ga) alloys.These alloys display metallic nature in one spin channel and semiconducting in other spins channel.V 2 MnAs show half-metallic nature in the spin-up channel, while V 2 MnGa shows half-metallic nature in spin-down channel.Using Birch Murnaghan equation [25], the equilibrium lattice constant is calculated.Here, B 0 is the bulk modulus, B' is its derivative, V 0 is the volume at zero pressure.The calculated optimized values are shown in table 1.

Electronic and magnetic property
In this section we have discussed about the electronic structure of V 2 MnZ (Z = As, Ga) Heusler alloys.We investigated the total density of states (TDOS), partial density of states (PDOS) and band structure of V 2 MnZ (Z = As, Ga) alloys respectively, its main aim is to explain the half metallic behaviour of these alloys.The total DOS plots of V 2 MnZ (Z = As, Ga) are shown in figures 4 and 5.No band gap is shown in spin-down region of   V 2 MnAs and in spin-up region of V 2 MnGa.On the other hand a clear band gap is seen in spin up region of V 2 MnAs and in spin-down region of V 2 MnGa between the conduction and the valence band.As a result, due to the exchange splitting the band gap arises between valence and conduction band in both the materials and thus reveals their half-metallic nature.Partial DOS (PDOS) plots of V 2 MnZ (Z = As, Ga) alloys are shown in figures 6 and 7.In V 2 MnZ (Z = As, Ga), V and Mn are the transition metals with two incomplete d-orbitals.Hence, they participate in d-d  Table 1.Represents calculated lattice parameter (a 0 in nm), magnetic moments (M in μ B ), volume (V 0 in nm 3 ), bulk modulus (B 0 in GPa), derivative of bulk modulus (B' 0 ), energy (E 0 in eV) and number of valence electrons (Nv).Our calculations confirmed that V 2 MnZ (As, Ga) are half-metallic ferromagnets with total spin magnetic moments of 4 μ B and 2 μ B respectively with 100% spin polarization.For Heusler alloys, the magnetic moment values are consistent with the Slater-Pauling rule, M t = Z t −18, where M t denotes the total magnetic moment and Z t denotes the total no of valence electrons.Since the Z elements belong to different groups and the number of electrons in the valence band is different, the calculated value of the magnetic moment is additional for other elements, which nicely matches the magnetic moments predicted with the Slater-Pauling rule and shows that they have the potential to be half-metallic.Also, most of the contribution in magnetic moment is from d eg and d t2g states of V and Mn atom.

Thermodynamic properties
In order to explain the thermodynamic stability of these vanadium based V 2 MnZ (Z = As, Ga) Heusler alloys, we have used a quasi-harmonic approximation [28] of the Debye model (QHM), which significantly describes the stability of these various quantities, including the specific heat at a constant volume (Cv), the Gruneisen parameter (γ), and the thermal expansion coefficient (α), in the range of '0-900' K and '0-20' GPa, respectively.The material's capability to absorb heat from its surroundings is called heat Capacity.C V tends towards a constant value at high temperatures, displaying the Dulong and Petit law.However, C V changes abruptly at lower temperatures, following Debye's law T 3 .It maintains a sharply rising trend at lower temperatures while maintaining a constant value at higher temperatures.Figure 12 shows the variation of heat capacity (C v ) with  temperature for V 2 MnZ (Z = As, Ga).The obtained heat capacity value for V 2 MnAs is 74.74Jmol −1 K −1 , and V 2 MnGa is.74.75 Jmol −1 K −1 respectively.
In thermodynamic properties, the Grüneisen parameter (g ) explains the correlation between the change in the volume of crystals and the frequency of phonons.It is also a significant dynamical property used to quantify the relationship between elastic and thermal descriptions of solids and is directly connected to these two.Figure 13 shows the change in the Grüneisen parameter (g ) with temperature for V 2 MnZ (Z = As, Ga).The obtained value of the Grüneisen parameter at zero pressure and 300 K for V 2 MnAs is 1.978, and for V 2 MnGa is 1.976.Thermal expansion (α) refers to the tendency of matter to change its shape, area, volume, and density in response to a change in temperature.Figure 14 shows variation in thermal expansion with temperature.The obtained thermal expansion value at zero pressure and 300 K for V 2 MnZ (Z = As, Ga) are 1.33 and 1.34, respectively.Debye temperature (θ D ) is the temperature at which the collective (acoustic) variation changes to  free thermal vibrations.Figure 15 shows the plot of Debye temperature (θ D ) versus temperature.It is also clear from the figure that the Debye temperature has a constant value of 754.25 K up to 50 K approximately for V 2 MnAs, and for V 2 MnGa, it is 754.24K and beyond this, as the temperature increases, θ D decreases for both the alloys.

Cohesive energy
Cohesive energy is one of the essential thermodynamic properties that determine materials' strength and stability.Cohesive energy [29] is required to dissociate or liberate atoms within the crystal lattice.The calculated cohesive energy of the alloys is 5.67 eV atom −1 for V 2 MnAs and 5.58 eV atom for V 2 MnGa, respectively.The following equation is used to compute cohesive energy (E Coh ):

Thermoelectric property
Currently, fossil fuels are mainly responsible for generating energy.Therefore, efficient, easily manipulatable, and eco-friendly substitutes are needed.(TE) materials are those which can increase engine operating efficiency  by converting waste energy into electricity efficiently.The thermoelectric properties were calculated using constant relaxation time approx.whose value is τ = 0.5 * 10 -14 s. on which Boltztrap code is based.The thermoelectric properties provide an essential understanding of charge transfer, which determines a material's thermoelectric response to renewable energy sources.Thermoelectric materials [30] are electronic materials used for cooling and power generation.The efficiency of thermoelectric is based on three thermodynamic impacts, the Seebeck, Peltier [31], and Thomson impact.In the Seebeck effect, an electrical potential gradient is generated when a temperature gradient is applied, which is used for power generation.These materials are good for thermoelectric devices as they have high electrical and low thermal conductivity and a significant Seebeck coefficient.In both spin channels, we assessed the different transport coefficients such as electrical conductivity, electronic thermal conductivity, Seebeck coefficient, power factor, and dimensionless figure of merit (ZT) for the V 2 MnZ (Z = As, Ga) Heusler alloys.

Seebeck coefficient (S)
Seebeck coefficient [32] is used to determine the magnitude of the induced thermoelectric voltage as a function of the temperature difference across a material, as measured by the Seebeck coefficient.In general, materials with a higher Seebeck coefficient are more thermoelectrically conductive.It is evident from the negative and positive values of the total Seebeck coefficient that Heusler consist of p-type and n-type semiconductors, depending on holes and electrons as their suppliers.According to the two-current model [33], the total Seebeck coefficient (S) can be expressed as the average of the Seebeck coefficients from both channels weighted by the conductivities of the two channels: Figures 16 and 17 shows the variation of Seebeck coefficient as a function of temperature of V 2 MnZ (Z = As, Ga).The negative value obtained for Seebeck coefficient in the spin-up state of V 2 MnAs and the spin-dn state of V 2 MnGa alloys indicates that the electrons are highly dominant in these channels.The variation of Seebeck coefficient with temperature in the spin-up state of V 2 MnAs channel shows decreasing trend from 0.83 μV K −1 to −17.49 μV K −1 and in spin-down state of V 2 MnGa the decreasing trend is from 0.37 to −43.82 μV K −1 .Whereas, the positive value obtained for Seebeck coefficient in the spin-dn state of V2MnAs and the spin-up state of V 2 MnGa alloys indicates that the holes are the main charge carriers.The variation of Seebeck coefficient with temperature shows the increasing trend from 4.17 μV K −1 to 36.22 μV K −1 in the spin -dn state of V 2 MnAs while in the spin-up state of V 2 MnGa shows increasing trend from 4.57 μV K −1 to 62.27 μV K −1 .The decreasing trend of Seebeck coefficient also represents the metallic nature of the alloys while increasing value of Seebeck coefficient represents the semiconducting nature of these alloys.The Seebeck coefficient is expressed by following relation: - Here, 'm * ' represents the effective mass, 'n' refers to the carrier concentrations, and 'T' represents absolute temperature, K B is Boltzmann constant

Electrical conductivity (σ)
Electrical conductivity is an important property of electronic devices.It is possible to categorize materials into metals, ceramics, semiconductors, etc Electrical conductivity in these materials differs from each other due to the width of fermi level.In metals, free electrons as well as narrow band-gaps are present, whereas in semiconductors, the band-gap is much smaller.Figure 18 shows the deviation of electrical conductivity with temperature.When the temperature rises, an inclination trend can be easily observed in electrical conductivity  This is because as there is an increase in the carrier concentration, the conductivity(σ) of the materials also increases.The carrier concentration and conductivity are interrelated to each other by a mathematical equation ne , s m = where μ is the mobility.In V 2 MnAs, the electrical conductivity increases in the spin-up state from 2.69 * 10 20 (Ω −1 m −1 s −1 ) at 50 K to 2.43 * 10 20 (Ω −1 m −1 s −1 ) at 900 K and shows decreasing trend in spin down state from 2.69 * 10 20 (Ω −1 m −1 s −1 ) at 50 K to 2.43 * 10 20 (Ω −1 m −1 s −1 ) while in V 2 MnGa conductivity increases in spin-down state from 0.78 * 10 20 (Ω −1 m −1 s −1 ) at 50 K to 0.91 * 10 20 (Ω −1 m −1 s −1 ) at 900 K and shows decreasing trend in spin-up channel from 1.95 * 10 20 (Ω −1 m −1 s −1 ) at 50 K to 1.76 * 10 20 (Ω −1 m −1 s −1 ) at 900 K. respectively.Total electrical conductivity is represented as follows: Where, s  and s  are the conductivities in spin up and spin down channels.

Total thermal conductivity(κ)
Electronic and lattice components contribute to thermal conductivity.The thermal conductivity κ has mainly two components: Where κ e is electronic conductivity and κ l is lattice conductivity.According to the Weidman-Franz law [34], higher the electrical conductivity, higher will be .where e k  and e k  are the conductivities in spin-up and spin-down channels.The lattice thermal conductivity can be calculated by the Slack model, which relates to the Debye temperature obtained from the elastic constants.Through Slacks equation [35] κ ℓ . is well-defined: Here, A is constant whose values is 3.04 * 10 −8 , n is the number of atoms in the primitive unit cell, θ D is the Debye temperature, V is the average volume, M is the average molar mass, T is the temperature, and γ is the Grüneisen parameter.Lattice thermal conductivity is usually calculated using thermodynamic parameters.Figures 19 and  20 shows the variation of total thermal conductivity as a function of temperature respectively.Lattice thermal conductivity almost becomes zero as the temperature rises while electronic conductivity rises with rise in  temperature and total thermal conductivity increase with increase in temperature for V 2 MnZ (Z = As, Ga) Heusler alloys.

Power factor
The power factor of a material can be used to find the thermoelectric efficiency.Power factor (PF) dictates a material's thermo-electrical response.Power factor, which may be expressed as S 2 σ, the product of the Seebeck coefficient (S) and electrical conductivity (σ).S and σ should both be high in order to attain a high-power factor.Figure 21 demonstrates variation of power factor of V 2 MnZ (Z = As, Ga) alloy with temperature, which shows a linear increase with temperature.By measuring the power factor, we can categorize materials based on their output generated at the extent of a temperature gradient.The calculated power factor for V 2 MnAs at 300 K is 3.24 × 10 −3 W mK −2 , and for V 2 MnGa is 0.27 × 10 −3 W mK −2 .The power factor is indicated as follows: To obtain a high ZT, the material must be thermoelectrically stable, it should have a high Seebeck coefficient (S), high electrical conductivity(σ), and a low thermal conductivity(κ).Figure 22 shows the variation of figure of merit as a function of temperature for V 2 MnZ (Z = As, Ga) Heusler alloys.The obtained figure 20 of merit (ZT) for V 2 MnAs at 300 K is 0.076 and at 800 K is 0.66 while for V 2 MnGa at 300 K is 0.009 and at 800 K is 0.65 respectively.

Conclusions
We investigated the electro-magnetic, thermophysical, and thermoelectric properties of V 2 MnAs and V 2 MnGa Heusler alloys by using first principle calculations.Geometrical stability of the alloys has been demonstrated in cubic phase F-43m by structural optimization.For V 2 MnZ (Z = As, Ga), the spin magnetic moment is 4 μB and 2 μB respectively.The alloys are thermodynamically stable.The predicted Seebeck coefficients are sufficiently large to indicate that these alloys can be used as thermoelectric materials.The important ZT measurement favors its use in high-temperature thermoelectric technology with a highly significant density of states around the Fermi level.Therefore, these alloys have a wide variety of potential applications in energy harvesting technologies and other research areas that need to be explored further.

Funding
There is no funding for the research.

Figure 2 .
Figure 2. Variation of Energy (eV) as a function of volume (nm 3 ) of V 2 MnAs in FM and NM phases.

Figure 3 .
Figure 3. Variation of Energy (eV) as a function of volume (nm 3 ) of V 2 MnGa in FM and NM phases.

Figure 4 .
Figure 4. Graphical representation of TDOS of V 2 MnAs in GGA and mBJ approximations.

Figure 5 .
Figure 5. Graphical representation of TDOS of V 2 MnGa in GGA and mBJ approximations.

Figure 6 .
Figure 6.Graphical representation of PDOS of V 2 MnAs in spin-up and spin-down state.

Figure 7 .
Figure 7. Graphical representation of PDOS of V 2 MnGa in spin-up and spin-down state.

Figure 8 .
Figure 8. Band plots of V 2 MnAs alloy in GGA approximations in spin-up and spin-down channel.

Figure 9 .
Figure 9. Band plots of V 2 MnAs alloy in mBJ approximations in spin-up and spin-down channel.

Figure 10 .
Figure 10.Band plots of V 2 MnGa alloy in GGA approximations in spin-up and spin-down channel.

Figure 11 .
Figure 11.Band plots of V 2 MnGa alloy in mBJ approximations in spin-up and spin-down channel.

Figure 12 .
Figure 12.Variation of heat capacity (C V ) as a function of temperature for V 2 MnZ (Z = As, Ga).

Figure 13 .
Figure 13.Variation of grüneisen parameter (γ) as a function of temperature for V 2 Mn (Z = As, Ga).

Figure 14 .
Figure 14.Variation of thermal expansion coefficient (α) as a function of temperature for V 2 MnZ (Z = As, Ga).

Figure 15 .
Figure 15.Variation of Debye temperature (θ D ) as a function of temperature for V 2 MnZ (Z = As, Ga).

Figure 16 .
Figure 16.Variation of seebeck coefficient (S) as a function of temperature for V 2 MnAs.

Figure 17 .
Figure 17.Variation of seebeck coefficient (S) as a function of temperature for V 2 MnGa.

Figure 19 .
Figure 19.Variation of thermal conductivity as a function of temperature for V 2 MnAs.

Figure 18 .
Figure 18.Variation of electrical conductivity(σ) as a function of temperature for V 2 MnZ (Z = As, Ga).

ek
The electronic conductivity is expressed as:

Figure 20 .
Figure 20.Variation of thermal conductivity as a function of temperature for V 2 MnGa.

Figure 21 .
Figure 21.Variation of power factor (PF) as a function of temperature for V 2 MnZ (Z = As, Ga).

5 .
Figure of merit (ZT)Thermoelectric cooling requires a material with a high thermoelectric figure of merit (ZT)[36].The figure of merit is represented as follows:

Figure 22 .
Figure 22.Variation of figure of merit (ZT) as a function of temperature for V 2 MnZ (Z = As, Ga).